In how many of the arrangements would the two L's be together, for the word BASKETBALL?
The number of arrangements for n distinct lettes is
n!
For example: 4!=24 distinct words can be made from the letters ABCD.
The number of arrangements for n letters, of which p are identical is
n!/p!
For example: 4!/2!=12 words can be made from the letters AABC.
The number of arrangements for n letters, of which p are identical and q are identical is
n!/(p!q!)
For example: 6!/(2!2!)=180 words can be made from the letters AABBCD.
If two letters have to be together all the timee, treat them as a single letter.
For the word BASKETBALL,
first arrange them in alphabetical order:
AABBEKLLST
Out of the 10 letters, AA and BB are repetitions, LL can be treated as one single letter (to have a total of 11 letters).
So the number of distinct words possible is
11!/(...)
To find the number of arrangements where the two L's are together in the word "BASKETBALL," we can treat the two L's as a single entity. This way, we have nine entities to arrange: B, A, S, K, E, T, B, A, L(L).
To solve this problem, we can follow these steps:
Step 1: Count the total number of arrangements without any restrictions.
In this case, we have 9 entities to arrange. The total number of arrangements without any restrictions is given by the factorial of 9, which is denoted as 9!
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880
Step 2: Count the number of arrangements where the two L's are treated as a single entity.
Now, we consider the two L's as one entity in the arrangement, giving us 8 entities: B, A, S, K, E, T, B, A(L,L).
The total number of arrangements with the two L's treated as a single entity is given by the factorial of 8, which is denoted as 8!
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320
Step 3: Subtract the number of arrangements with the two L's treated as a single entity from the total number of arrangements without any restrictions.
362,880 - 40,320 = 322,560
Therefore, there are 322,560 arrangements of the word "BASKETBALL" where the two L's are together.